Answer
$\left[ -\dfrac{2}{3},4\right]$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
\left|5-3x \right|\le7
,$ remove the absolute value sign using the properties of absolute value inequality. Then use the properties of inequality to isolate the variable.
$\bf{\text{Solution Details:}}$
For any $a\gt0,$ $|x|\le a$ implies $-a\le x\le a.$ (Note that the symbol $\le$ may be replaced with $\lt.$) Hence, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-7\le5-3x \le7
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-7-5\le5-5-3x \le7-5
\\\\
-12\le-3x \le2
.\end{array}
Dividing both sides by $
-3
$ and reversing the inequality symbols, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{-12}{-3}\ge\dfrac{-3x}{-3} \ge\dfrac{2}{-3}
\\\\
4\ge x \ge-\dfrac{2}{3}
\\\\
-\dfrac{2}{3}\le x \le4
.\end{array}
Hence, the solution is the interval $
\left[ -\dfrac{2}{3},4\right]
.$
